Series solution of Laplace problems

TL;DR

This paper introduces a series expansion method for solving Laplace problems in multiply connected domains, offering an elementary alternative to conformal mapping with exponential convergence under certain conditions.

Contribution

It presents a tutorial on a series expansion approach for Laplace problems, highlighting its convergence properties and historical mathematical foundations.

Findings

Method converges exponentially with well-behaved boundary data.

Applicable to complex domains, including those with many components.

Provides an accessible alternative to conformal mapping techniques.

Abstract

At the ANZIAM conference in Hobart in February, 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.